Group Homomorphisms: Definitions & Sample Calculations

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  • 0:04 Special Functions in Math
  • 0:42 What is Group Homomorphism?
  • 2:41 Determining a Homomorphism
  • 4:26 Why are Homomorphisms Useful?
  • 5:37 Lesson Summary
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Lesson Transcript
Instructor: Michael Gundlach
We're going to talk about a special type of function on a group called a homomorphism. This special function not only maps elements from one group to another, it actually preserves some properties of the group as we map from one to another.

Special Functions in Math

In high school or college, we often learn about special functions called logarithms and exponents. As part of learning about these functions, we learned exponent laws and logarithm properties like the following:

2b + c = 2b · 2c

and

log(x · y) = log(x) + log(y).

For both functions, an operation between two inputs of the function resulted in an operation between the two outputs of these functions. This preservation of operations means that both exponents and logarithms are special functions called group homomorphisms.

What is Group Homomorphism?

Before jumping in and defining a group homomorphism, remember that we often represent a group using the notation (G, ∗), where G is the set and ∗ is the operation. So, for example, (R, +) is the group of real numbers under addition. Sometimes, if the operation is known or clear from context, we'll abbreviate by just calling the group G. For example, when we talk about the group of integers under addition, we usually just write Z since the integers do not form a group under any of the other basic operations.

A group homomorphism (often just called a homomorphism for short) is a function ƒ from a group (G, ∗) to a group (H, ◊) with the special property that for a and b in G,

ƒ(a ∗ b) = ƒ(a) ◊ ƒ(b)

Let's look at some examples to help make this more clear. First, let's look back at the first example. The function ƒ(x) = 2x can be considered a function from (R, +) to (R+, ·), where R+ is the set of all positive real numbers. We can tell it's a homomorphism since:

ƒ(b + c) = 2b + c = 2b · 2c = ƒ(b) · ƒ(c).

Another homomorphism that might be familiar is the map φ from Z to Z7 (the group of integers modulo 7 under addition) given by φ(x) =[x], where [x] represents x mod 7. Remember that x mod 7 is the distance of x from a multiple of 7. So, for example, 65 is 2 more than 63 (which is 7 · 9), meaning 65 mod 7 can be represented by [2]. We can see this is a homomorphism since we know from modular arithmetic that

φ(x+y) = [x + y] = [x] + [y] = φ(x) + φ(y).

Determining a Homomorphism

A common problem when working in abstract algebra is to determine if a certain function is a homomorphism. To determine if a function is a homomorphism, we simply need to check that the function preserves the operation. In other words, we need to make sure that for a function ƒ from a group (G, ∗) to a group (H, ◊) that

ƒ(a ∗ b) = ƒ(a) ◊ ƒ(b)

is true for all a and b in G. We often can't check all of the pairs of elements in G (since it could be infinitely large). Therefore, we check to see if the above property holds for arbitrary a and b.

For example, let's determine if the function ƒ from (R, +) to (R, +) given by ƒ(x)= 5x is a homomorphism. Since there is an infinite amount of real numbers, we're going to use x and y to represent arbitrary real numbers. Let's see if we can get the homomorphism property to work.

ƒ(x + y) = 5(x + y) = 5x + 5y = ƒ(x) + ƒ(y).

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